forked from 1b-agent/fieldcraft-upstream
jyvikoandClaude Opus 5.5
3f9e9e30d8
Temperatures now strain the material in nonlinear steps instead of being refused. At finite strain the deformation gradient splits multiplicatively, F = F_m F_theta with F_theta = diag(1 + alpha_i dT) at each point: the material sees F_m and the loops run on the thermally expanded volume J_theta dV0 (mixed elements project det F_m), so the tangent stays consistent. At small strain the split is additive, and elements on their linear stiffness take the equivalent nodal forces as a dead load. The step's temperatures are its temperature loads, or the steady conduction of its temperature conditions and heat loads (one-way coupling, as the linear coupled step); both at once are refused. They ramp with the load factor from each material's reference temperature. Johnson-Cook plasticity softens at the point's temperature. Verified: free expansion and a fully constrained neo-Hookean block at 20 % thermal stretch, full and hybrid (exact); a bar held at its ends heated past yield (exact); the linear limit and the conducted temperatures against the linear solvers; tangents by central differences with a temperature field. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm
1110 lines
54 KiB
Rust
1110 lines
54 KiB
Rust
//! Nonlinear static analysis: equilibrium followed in load increments with Newton–Raphson
|
||
//! iterations.
|
||
//!
|
||
//! **Load path.** The step's load factor λ runs from 0 to 1. The dead loads (forces, tractions,
|
||
//! gravity, body forces; pressure too without nonlinear geometry) are λ times their values, a
|
||
//! follower pressure λ p on the deformed faces, and every prescribed displacement λ times its
|
||
//! value (u = T_f q + λ T_p u_p, the constraint elimination of the linear solver,
|
||
//! [`crate::constraints`]).
|
||
//!
|
||
//! **Newton–Raphson** with the full consistent tangent, refactored every iteration (Bathe
|
||
//! 2014, §8.4; M. A. Crisfield, *Non-linear Finite Element Analysis of Solids and Structures*,
|
||
//! vol. 1, Wiley 1991, ch. 9): K_T δq = r with r = T_fᵀ (λ f_ext(u) − f_int(u)), then, when the
|
||
//! step asks for it (it is off by default: on large elastic rotations the full Newton step
|
||
//! converges in fewer iterations), a line search on the work of the residual along δq, G(s) = δq · r(q + s δq): s = 1 is taken when
|
||
//! |G(1)| ≤ 0.8 |G(0)| (always near the solution, so quadratic convergence is kept), otherwise
|
||
//! s is moved to the root of the line through (0, G(0)) and (s, G(s)), within [0.1, 1], at most
|
||
//! five times (Crisfield 1991, §9.3). An increment has converged when, at once,
|
||
//! - **force**: the largest residual component is at most `force` times the largest force the
|
||
//! step has seen (applied, internal, reaction), never less than a round-off floor of 10⁻⁸ of a
|
||
//! typical diagonal stiffness times the model's size;
|
||
//! - **displacement**: the largest correction is at most `displacement` times the largest
|
||
//! displacement of the increment;
|
||
//! - **energy**: the work of the correction against the residual, s δq · r, is at most `energy`
|
||
//! times that of the first iteration (K.-J. Bathe, A. P. Cimento, *Some practical procedures
|
||
//! for the solution of nonlinear finite element equations*, CMAME 22:59–85, 1980);
|
||
//!
|
||
//! or when the force residual is a thousandth of its tolerance (round-off: no correction can
|
||
//! improve it). With path-dependent materials, an increment that starts within the force
|
||
//! tolerance (a predictor that already solved it) waives the energy criterion, whose reference
|
||
//! would be round-off.
|
||
//!
|
||
//! **Automatic incrementation.** Each increment starts from the linear extrapolation of the
|
||
//! last converged one (the free motion scaled by the ratio of their sizes), which keeps the
|
||
//! inner nodes in step with the prescribed ones; from the last converged state when the
|
||
//! extrapolation inverts an element. The first increment is the step's `initial_increment`; two
|
||
//! increments in a row that converge in at most four iterations let the next grow by half (up
|
||
//! to `max_increment`), one that needs more than eight shrinks it by a quarter. An increment is
|
||
//! cut back to a quarter of its size when it fails: an element inverts, the residual is not
|
||
//! finite, the residual grows in two successive iterations, the tangent is not positive definite
|
||
//! (a limit point or an unstable material), or it does not converge in `max_iterations`. The
|
||
//! step fails cleanly, naming the load factor and the last reason, when an increment falls below
|
||
//! `min_increment`, when it has taken `max_increments` increments, or when the Newton iterations
|
||
//! of the whole step reach [`crate::Limits::max_newton_iterations`].
|
||
//!
|
||
//! **Follower pressure** and the Cholesky factorization: the load stiffness of a pressure on the
|
||
//! deformed surface is not symmetric (Schweizerhof & Ramm 1984); its skew part is left out of the
|
||
//! tangent (the symmetric part is kept), so the factorization stays the supernodal Cholesky of the
|
||
//! linear solver. The residual is always exact, so the converged solution is unchanged; only the
|
||
//! rate of convergence can drop where the skew part matters. Over a closed surface, or one whose
|
||
//! boundary is held, the skew parts of neighbouring faces cancel and nothing is left out; the
|
||
//! result reports the skew part's size relative to the load stiffness when it is not small.
|
||
//!
|
||
//! **Elements.** Solids (tet4, tet10, wedge6, hex8, hex20 full and reduced, hybrid hex8, hex20
|
||
//! and tet10) and 2D plane strain and axisymmetric elements, total Lagrangian
|
||
//! ([`fieldcraft_fem::finite`]); without nonlinear geometry an element of a plastic material
|
||
//! takes the same loops with small strain, and an elastic one its linear stiffness (K u).
|
||
//! Springs act along their fixed directions. Trusses, shells and beams (large rotations), plane
|
||
//! stress, incompatible modes (with nonlinear geometry, or with an inelastic material) and
|
||
//! centrifugal loads are refused by name.
|
||
//!
|
||
//! **Thermal strain.** The step's temperatures are its temperature loads, or, when it has
|
||
//! temperature boundary conditions or heat loads, their steady conduction solved first (one-way
|
||
//! coupling, as the linear coupled temperature–displacement step). They ramp with the load
|
||
//! factor from each material's reference temperature, ΔT(λ) = λ (T − T_ref). With nonlinear
|
||
//! geometry the deformation gradient splits multiplicatively, F = F_m F_θ with
|
||
//! F_θ = diag(1 + αᵢ ΔT), the material sees F_m and the stress is J_θ τ_m per reference volume
|
||
//! (fem's `Temperature`); with small strain the split is additive, ε_m = ε − α ΔT; elements on
|
||
//! their linear stiffness take the equivalent nodal forces, a dead load. A Johnson–Cook material
|
||
//! softens at the point's temperature.
|
||
//!
|
||
//! **Plasticity** (`Material::plastic`, J2 with the radial return explicit dynamics uses,
|
||
//! [`fieldcraft_fem::elastoplastic`]): Simo's exponential return map at finite strain, the
|
||
//! additive radial return at small strain, both with their consistent tangents, so Newton keeps
|
||
//! its quadratic rate. The plastic history is committed when an increment converges and
|
||
//! discarded when it is cut back. A static step has no time, so the Johnson–Cook strain-rate term
|
||
//! is refused (C = 0 is the rate-independent law), and kinematic hardening at finite strain is
|
||
//! refused (its back stress breaks the principal-frame return); damage, erosion and adiabatic
|
||
//! heating act in explicit dynamics only, with a warning. Results add the equivalent plastic
|
||
//! strain at the integration points.
|
||
//! Couplings stay the linear (small-rotation) constraints of the linear solver, with a warning.
|
||
//!
|
||
//! Element forces and tangents are computed in parallel and added in label order, so results are
|
||
//! the same bit for bit on any number of threads.
|
||
|
||
use fieldcraft_fem::constitutive::MaterialModel;
|
||
use fieldcraft_fem::dense::Matrix;
|
||
use fieldcraft_fem::element::{Extrapolation, extrapolation};
|
||
use fieldcraft_fem::finite::{Kinematics, Temperature};
|
||
use fieldcraft_fem::plane::Theory;
|
||
use fieldcraft_fem::quadrature::Rule;
|
||
use fieldcraft_fem::{DofKind, DofSet, ElementType, PointStress, face, von_mises};
|
||
use fieldcraft_model::{BoundaryCondition, Extent, Hardening, Load, Model, NonlinearStatic, Plastic, Procedure, SectionKind, Step};
|
||
use fieldcraft_parallel as parallel;
|
||
use serde::Serialize;
|
||
|
||
use crate::analysis::{
|
||
ElementStress, GRAIN, PointResult, ROUND, Setup, StaticResult, Stats, budget, factor_error, factor_raw, ignored_dashpots, pattern, reduction,
|
||
split,
|
||
};
|
||
use crate::cholesky::FactorError;
|
||
use crate::constraints::Reduction;
|
||
use crate::sparse::{Csr, add_block};
|
||
use crate::system::{Mesh, add_element_thermal_loads, loads, temperature_loads, translations};
|
||
use crate::{Error, Limits, Result};
|
||
|
||
/// One Newton iteration of an increment.
|
||
#[derive(Debug, Clone, Copy, PartialEq, Serialize)]
|
||
pub struct Iteration {
|
||
/// Largest residual component over the force scale.
|
||
pub force: f64,
|
||
/// Largest correction over the largest displacement of the increment.
|
||
pub displacement: f64,
|
||
/// Work of the correction against the residual over the first iteration's.
|
||
pub energy: f64,
|
||
/// The line search's step along the correction (1: the full Newton step).
|
||
pub line_search: f64,
|
||
}
|
||
|
||
/// One increment (converged, or cut back).
|
||
#[derive(Debug, Clone, PartialEq, Serialize)]
|
||
pub struct Increment {
|
||
/// From 1, attempts included.
|
||
pub number: usize,
|
||
/// The load factor reached (when converged) or attempted.
|
||
pub load_factor: f64,
|
||
/// Its size, as a fraction of the step.
|
||
pub size: f64,
|
||
pub iterations: Vec<Iteration>,
|
||
pub converged: bool,
|
||
/// Why it was cut back.
|
||
#[serde(skip_serializing_if = "Option::is_none")]
|
||
pub cutback: Option<String>,
|
||
}
|
||
|
||
/// The displacements at the end of a converged increment.
|
||
#[derive(Debug, Clone, PartialEq)]
|
||
pub struct Frame {
|
||
pub load_factor: f64,
|
||
/// Per node, in [`StaticResult::nodes`] order.
|
||
pub displacement: Vec<[f64; 3]>,
|
||
}
|
||
|
||
/// The increments of a nonlinear static step and the displacements after each converged one.
|
||
#[derive(Debug, Clone, PartialEq, Default)]
|
||
pub struct History {
|
||
pub increments: Vec<Increment>,
|
||
/// The undeformed state (load factor 0), then one per converged increment while they fit
|
||
/// [`MAX_FRAME_VALUES`].
|
||
pub frames: Vec<Frame>,
|
||
/// Newton iterations over the step.
|
||
pub iterations: usize,
|
||
}
|
||
|
||
impl History {
|
||
/// The load factor of every frame.
|
||
pub fn load_factors(&self) -> Vec<f64> {
|
||
self.frames.iter().map(|f| f.load_factor).collect()
|
||
}
|
||
}
|
||
|
||
/// Most displacement values the frames of one step keep (3 per node per frame).
|
||
pub const MAX_FRAME_VALUES: usize = 30_000_000;
|
||
|
||
/// Line search: accept a step whose residual work is at most this fraction of the first.
|
||
const LINE_SEARCH_ACCEPT: f64 = 0.8;
|
||
/// Line search: most trial steps, and the range of the step.
|
||
const LINE_SEARCH_TRIALS: usize = 5;
|
||
const LINE_SEARCH_MIN: f64 = 0.1;
|
||
/// Cutback factor of a failed increment, growth after fast ones and shrinking after slow ones.
|
||
const CUTBACK: f64 = 0.25;
|
||
const GROWTH: f64 = 1.5;
|
||
const SHRINK: f64 = 0.75;
|
||
const FAST_ITERATIONS: usize = 4;
|
||
const SLOW_ITERATIONS: usize = 8;
|
||
/// A force residual this fraction of its tolerance is round-off.
|
||
const ROUND_OFF: f64 = 1e-3;
|
||
|
||
/// A pressure on element faces (or 2D edges) that follows the deformation.
|
||
struct Follower {
|
||
element: u32,
|
||
/// Layout slots of the face's nodes, and their reference positions.
|
||
nodes: Vec<usize>,
|
||
xyz: Vec<[f64; 3]>,
|
||
magnitude: f64,
|
||
/// The out-of-plane extent of a 2D edge; `None` for a solid face.
|
||
extent: Option<Extent>,
|
||
/// The translations of its nodes (x, y, z, or x, y per node).
|
||
dofs: Vec<usize>,
|
||
}
|
||
|
||
/// Everything an evaluation of the out-of-balance forces needs.
|
||
struct Problem<'a> {
|
||
mesh: &'a Mesh,
|
||
red: &'a Reduction,
|
||
/// Dead loads at λ = 1.
|
||
dead: Vec<f64>,
|
||
followers: Vec<Follower>,
|
||
/// Springs (linear along their fixed directions), on the pattern.
|
||
springs: Option<Csr>,
|
||
/// Large-deformation kinematics.
|
||
finite: bool,
|
||
/// Each element's global degrees of freedom, in its matrices' order, and the kinds per node.
|
||
element_dofs: Vec<Vec<usize>>,
|
||
element_kinds: Vec<DofSet>,
|
||
/// Each element's material model.
|
||
materials: Vec<MaterialModel>,
|
||
/// The step's temperature field at λ = 1, per mesh node (`None` where the field leaves a
|
||
/// node at its material's reference temperature); `None` without one.
|
||
field: Option<Vec<Option<f64>>>,
|
||
/// The tangent's pattern, zero.
|
||
pattern: Csr,
|
||
}
|
||
|
||
/// The internal and external forces at a state, and the tangent when asked for.
|
||
struct Evaluation {
|
||
internal: Vec<f64>,
|
||
external: Vec<f64>,
|
||
tangent: Option<Csr>,
|
||
/// Every element's material history at this state (committed when the increment
|
||
/// converges).
|
||
history: Vec<Vec<f64>>,
|
||
}
|
||
|
||
/// Why an attempt at an increment failed, when that calls for a cutback rather than an error.
|
||
enum Failure {
|
||
Cutback(String),
|
||
Fatal(Error),
|
||
}
|
||
|
||
impl From<Error> for Failure {
|
||
fn from(e: Error) -> Failure {
|
||
match e {
|
||
// An inverted element or a material that cannot take the deformation: a smaller
|
||
// increment may avoid it.
|
||
Error::Element { label, source: fieldcraft_fem::Error::Degenerate(m) } => Failure::Cutback(format!("element {label}: {m}")),
|
||
Error::NotConverged(m) => Failure::Cutback(m),
|
||
e => Failure::Fatal(e),
|
||
}
|
||
}
|
||
}
|
||
|
||
fn norm_inf(v: &[f64]) -> f64 {
|
||
v.iter().fold(0.0, |m, x| m.max(x.abs()))
|
||
}
|
||
|
||
fn dot(a: &[f64], b: &[f64]) -> f64 {
|
||
a.iter().zip(b).map(|(x, y)| x * y).sum()
|
||
}
|
||
|
||
/// ½(K + sign Kᵀ) of a row-major m × m matrix: its symmetric (sign 1) or skew (−1) part.
|
||
fn half(k: &[f64], m: usize, sign: f64) -> Vec<f64> {
|
||
(0..m * m).map(|i| 0.5 * (k.get(i).copied().unwrap_or(0.0) + sign * k.get((i % m) * m + i / m).copied().unwrap_or(0.0))).collect()
|
||
}
|
||
|
||
/// The out-of-balance forces over the free equations, T_fᵀ (f_ext − f_int).
|
||
fn free_residual(red: &Reduction, e: &Evaluation) -> Vec<f64> {
|
||
let r: Vec<f64> = e.external.iter().zip(&e.internal).map(|(a, b)| a - b).collect();
|
||
red.restrict(&r)
|
||
}
|
||
|
||
impl Problem<'_> {
|
||
/// Nodal displacements of an element (3 per node for solids, 2 for 2D elements).
|
||
fn element_vector(&self, e: usize, u: &[f64]) -> Vec<f64> {
|
||
self.element_dofs.get(e).map(|dofs| dofs.iter().map(|d| u.get(*d).copied().unwrap_or(0.0)).collect()).unwrap_or_default()
|
||
}
|
||
|
||
/// How element `e` is evaluated: total Lagrangian with nonlinear geometry, with small
|
||
/// strain when its material is inelastic, and by its linear stiffness otherwise (`None`).
|
||
fn kinematics(&self, e: usize) -> Option<Kinematics> {
|
||
if self.finite { Some(Kinematics::Finite) } else { self.materials.get(e).filter(|m| m.is_path_dependent()).map(|_| Kinematics::Small) }
|
||
}
|
||
|
||
/// The temperatures of element `e` at load factor λ, which ramps them from the reference
|
||
/// temperature to the field's (J2 Johnson–Cook softening and thermal strain): the
|
||
/// material's expansion and reference temperature, or no expansion from the Johnson–Cook
|
||
/// room temperature; `None` without a field or for a material neither affects.
|
||
fn temperature(&self, e: usize, lambda: f64) -> Option<Temperature> {
|
||
let field = self.field.as_ref()?;
|
||
let el = self.mesh.elements.get(e)?;
|
||
let softens = el.plastic.as_ref().is_some_and(|j2| j2.is_temperature_dependent());
|
||
let (alpha, reference) = match el.expansion {
|
||
Some(x) => x,
|
||
None if softens => ([0.0; 3], el.plastic.as_ref().map_or(0.0, |j2| j2.initial_temperature())),
|
||
None => return None,
|
||
};
|
||
let nodal: Vec<f64> =
|
||
el.nodes.iter().map(|n| field.get(*n).copied().flatten().map_or(reference, |t| reference + lambda * (t - reference))).collect();
|
||
(softens || nodal.iter().any(|t| *t != reference)).then_some(Temperature { alpha, reference, nodal })
|
||
}
|
||
|
||
/// Internal forces and the tangent of element `e` at displacements `u` (all degrees of
|
||
/// freedom) and load factor λ.
|
||
fn element(&self, e: usize, u: &[f64], lambda: f64, tangent: bool, committed: &[f64]) -> Result<(Vec<f64>, Option<Matrix>, Vec<f64>)> {
|
||
let el = self.mesh.elements.get(e).ok_or_else(|| Error::Internal("element index".into()))?;
|
||
let at = |err| Error::element(el.label, err);
|
||
let ue = self.element_vector(e, u);
|
||
let kernel = el.kernel()?;
|
||
let Some(kin) = self.kinematics(e) else {
|
||
let k = kernel.stiffness(&el.d).map_err(at)?;
|
||
let f = k.mul_vec(&ue);
|
||
return Ok((f, tangent.then_some(k), committed.to_vec()));
|
||
};
|
||
let material = self.materials.get(e).ok_or_else(|| Error::Internal("element material".into()))?;
|
||
let mut updated = committed.to_vec();
|
||
let r = kernel.nonlinear(material, &ue, kin, self.temperature(e, lambda).as_ref(), committed, &mut updated, tangent).map_err(at)?;
|
||
Ok((r.force, r.tangent, updated))
|
||
}
|
||
|
||
/// The forces of a follower pressure at λ (on [`Follower::dofs`]) and its load stiffness
|
||
/// (full, not symmetric).
|
||
fn follower(&self, f: &Follower, u: &[f64], lambda: f64) -> Result<(Vec<f64>, Vec<f64>)> {
|
||
let at = |err| Error::element(f.element, err);
|
||
let ue: Vec<f64> = f.dofs.iter().map(|d| u.get(*d).copied().unwrap_or(0.0)).collect();
|
||
let p = lambda * f.magnitude;
|
||
let (force, k) = match f.extent {
|
||
None => {
|
||
let (force, k) = face::follower_pressure(&f.xyz, ue.as_chunks::<3>().0, p).map_err(at)?;
|
||
(force.into_iter().flatten().collect(), k)
|
||
}
|
||
Some(extent) => {
|
||
let (force, k) = face::follower_edge_pressure(&f.xyz, ue.as_chunks::<2>().0, extent, p).map_err(at)?;
|
||
(force.into_iter().flatten().collect(), k)
|
||
}
|
||
};
|
||
Ok((force, k))
|
||
}
|
||
|
||
/// The forces at displacements `u` and load factor λ, and the tangent (symmetric part of the
|
||
/// load stiffness included) when `tangent`.
|
||
fn evaluate(&self, u: &[f64], lambda: f64, tangent: bool, committed: &[Vec<f64>]) -> Result<Evaluation> {
|
||
let layout = &self.mesh.layout;
|
||
let n = u.len();
|
||
let mut internal = vec![0.0; n];
|
||
let mut k = tangent.then(|| self.pattern.clone());
|
||
let mut history = Vec::with_capacity(self.mesh.elements.len());
|
||
for round in parallel::blocks(self.mesh.elements.len(), ROUND) {
|
||
let start = round.start;
|
||
let computed = parallel::map_grained(round.len(), GRAIN, |i| {
|
||
self.element(start + i, u, lambda, tangent, committed.get(start + i).map(Vec::as_slice).unwrap_or_default())
|
||
});
|
||
for (i, c) in computed.into_iter().enumerate() {
|
||
let (f, ke, updated) = c?;
|
||
history.push(updated);
|
||
let e = start + i;
|
||
let el = self.mesh.elements.get(e).ok_or_else(|| Error::Internal("element index".into()))?;
|
||
for (d, v) in self.element_dofs.get(e).into_iter().flatten().zip(&f) {
|
||
if let Some(x) = internal.get_mut(*d) {
|
||
*x += v;
|
||
}
|
||
}
|
||
if let (Some(k), Some(ke)) = (k.as_mut(), ke) {
|
||
let kinds = self.element_kinds.get(e).copied().ok_or_else(|| Error::Internal("element kinds".into()))?;
|
||
add_block(k, layout, &el.nodes, kinds, &ke.data)?;
|
||
}
|
||
}
|
||
}
|
||
if let Some(s) = &self.springs {
|
||
let mut ku = vec![0.0; s.n];
|
||
s.mul(u.get(..s.n).unwrap_or_default(), &mut ku);
|
||
for (x, v) in internal.iter_mut().zip(&ku) {
|
||
*x += v;
|
||
}
|
||
// Both are on the mesh's pattern.
|
||
if let Some(k) = k.as_mut() {
|
||
k.val.iter_mut().zip(&s.val).for_each(|(x, v)| *x += v);
|
||
}
|
||
}
|
||
let mut external: Vec<f64> = self.dead.iter().map(|f| lambda * f).collect();
|
||
for f in &self.followers {
|
||
let (force, kl) = self.follower(f, u, lambda)?;
|
||
for (d, v) in f.dofs.iter().zip(&force) {
|
||
if let Some(x) = external.get_mut(*d) {
|
||
*x += v;
|
||
}
|
||
}
|
||
if let Some(k) = k.as_mut() {
|
||
add_block(k, layout, &f.nodes, self.mesh.translation_kinds(), &half(&kl, f.dofs.len(), 1.0))?;
|
||
}
|
||
}
|
||
if internal.iter().chain(&external).any(|v| !v.is_finite()) {
|
||
return Err(Error::NotConverged("the forces are not finite".into()));
|
||
}
|
||
Ok(Evaluation { internal, external, tangent: k, history })
|
||
}
|
||
|
||
/// The size of the follower load stiffness's skew part relative to the whole, over the free
|
||
/// degrees of freedom (Frobenius norms), at `u` and λ; `None` without followers.
|
||
fn follower_asymmetry(&self, u: &[f64], lambda: f64) -> Result<Option<f64>> {
|
||
if self.followers.is_empty() {
|
||
return Ok(None);
|
||
}
|
||
let layout = &self.mesh.layout;
|
||
let (mut skew, mut whole) = (self.pattern.clone(), self.pattern.clone());
|
||
for f in &self.followers {
|
||
let (_, kl) = self.follower(f, u, lambda)?;
|
||
add_block(&mut skew, layout, &f.nodes, self.mesh.translation_kinds(), &half(&kl, f.dofs.len(), -1.0))?;
|
||
add_block(&mut whole, layout, &f.nodes, self.mesh.translation_kinds(), &kl)?;
|
||
}
|
||
let mut free = vec![false; skew.n];
|
||
for d in &self.red.eq_dof {
|
||
if let Some(x) = free.get_mut(*d) {
|
||
*x = true;
|
||
}
|
||
}
|
||
let norm = |a: &Csr| -> f64 {
|
||
let mut sum = 0.0;
|
||
for r in (0..a.n).filter(|r| free.get(*r).copied().unwrap_or(false)) {
|
||
let (cols, vals) = a.row(r);
|
||
for (c, v) in cols.iter().zip(vals) {
|
||
if free.get(*c).copied().unwrap_or(false) {
|
||
sum += v * v;
|
||
}
|
||
}
|
||
}
|
||
sum.sqrt()
|
||
};
|
||
let total = norm(&whole);
|
||
Ok((total > 0.0).then(|| norm(&skew) / total))
|
||
}
|
||
}
|
||
|
||
/// Mutable state of the Newton loop over the step.
|
||
struct Run<'a> {
|
||
problem: &'a Problem<'a>,
|
||
settings: &'a NonlinearStatic,
|
||
limits: &'a Limits,
|
||
/// The largest force seen so far (∞-norm), and the round-off floor of the force scale.
|
||
force_scale: f64,
|
||
force_floor: Option<f64>,
|
||
length: f64,
|
||
iterations: usize,
|
||
factor_entries: usize,
|
||
/// Every element's material history at the end of the last converged increment
|
||
/// (integration points × the material's history length).
|
||
committed: Vec<Vec<f64>>,
|
||
/// Whether a material keeps history (the tangent predictor and its convergence rules).
|
||
path_dependent: bool,
|
||
/// Whether the undeformed model's tangent factors, once asked.
|
||
undeformed_stable: Option<bool>,
|
||
}
|
||
|
||
/// The outcome of a converged increment.
|
||
struct Converged {
|
||
q: Vec<f64>,
|
||
evaluation: Evaluation,
|
||
}
|
||
|
||
impl Run<'_> {
|
||
/// The residual over the free equations, T_fᵀ (f_ext − f_int), and the force scale.
|
||
fn residual(&mut self, e: &Evaluation) -> Vec<f64> {
|
||
self.force_scale = self.force_scale.max(norm_inf(&e.external)).max(norm_inf(&e.internal));
|
||
free_residual(self.problem.red, e)
|
||
}
|
||
|
||
fn scale(&self) -> f64 {
|
||
self.force_scale.max(self.force_floor.unwrap_or(0.0)).max(f64::MIN_POSITIVE)
|
||
}
|
||
|
||
fn state(&self, q: &[f64], lambda: f64, tangent: bool) -> std::result::Result<Evaluation, Failure> {
|
||
Ok(self.problem.evaluate(&self.problem.red.expand_scaled(q, lambda), lambda, tangent, &self.committed)?)
|
||
}
|
||
|
||
/// Whether the tangent of the undeformed, unloaded model is positive definite: a first
|
||
/// tangent that is not, at a state the first increment has already strained, is then the
|
||
/// material or the geometry (a cutback), not a model that can move freely (an error).
|
||
fn undeformed_is_stable(&mut self) -> std::result::Result<bool, Failure> {
|
||
if let Some(stable) = self.undeformed_stable {
|
||
return Ok(stable);
|
||
}
|
||
let mut e = self.problem.evaluate(&vec![0.0; self.problem.mesh.dofs()], 0.0, true, &self.committed)?;
|
||
let k = e.tangent.take().ok_or_else(|| Failure::Fatal(Error::Internal("tangent".into())))?;
|
||
let [kff] = self.problem.red.reduce([&k])?;
|
||
let stable = factor_raw(&kff, self.limits).is_ok();
|
||
self.undeformed_stable = Some(stable);
|
||
Ok(stable)
|
||
}
|
||
|
||
/// The tangent predictor of an increment from the converged `q0` at load factor `from` to
|
||
/// `to`, and its work δq · r (the increment's reference for the energy criterion, since the
|
||
/// Newton iterations after it only correct it): one linear solve with the tangent of the converged state for the change of the
|
||
/// loads and of the prescribed displacements, K δq = T_fᵀ (f_ext(to) − f_int(u₀) − K T_p Δλ u_p).
|
||
/// It spreads a prescribed motion into the interior before the material sees it, which a
|
||
/// path-dependent material needs: started from the boundary's jump alone, the elements next
|
||
/// to it would flow plastically far beyond the solution. `None` when the tangent cannot be
|
||
/// factored (the increment then starts from `q0`).
|
||
fn tangent_predictor(&mut self, q0: &[f64], from: f64, to: f64) -> std::result::Result<Option<(Vec<f64>, f64)>, Failure> {
|
||
let red = self.problem.red;
|
||
let mut e = self.problem.evaluate(&red.expand_scaled(q0, from), to, true, &self.committed)?;
|
||
let k = e.tangent.take().ok_or_else(|| Failure::Fatal(Error::Internal("tangent".into())))?;
|
||
let jump = red.expand_scaled(&vec![0.0; red.neq()], to - from);
|
||
let mut kj = vec![0.0; jump.len()];
|
||
k.mul(&jump, &mut kj);
|
||
let rf: Vec<f64> = free_residual(red, &e).iter().zip(red.restrict(&kj)).map(|(a, b)| a - b).collect();
|
||
let mut dq = rf.clone();
|
||
let [kff] = red.reduce([&k])?;
|
||
drop(k);
|
||
let Ok(chol) = factor_raw(&kff, self.limits) else { return Ok(None) };
|
||
chol.solve(&mut dq)?;
|
||
let energy = dot(&dq, &rf).abs();
|
||
Ok(dq.iter().all(|v| v.is_finite()).then(|| (q0.iter().zip(&dq).map(|(a, d)| a + d).collect(), energy)))
|
||
}
|
||
|
||
/// Newton iterations from the converged `q0` (at the previous load factor) to load factor
|
||
/// `lambda`.
|
||
fn increment(
|
||
&mut self,
|
||
q0: &[f64],
|
||
predictor: Option<&[f64]>,
|
||
predicted_energy: Option<f64>,
|
||
lambda: f64,
|
||
size: f64,
|
||
iterations: &mut Vec<Iteration>,
|
||
) -> std::result::Result<Converged, Failure> {
|
||
let red = self.problem.red;
|
||
let tol = self.settings.tolerances;
|
||
// Start from the extrapolated predictor when it is a valid state, else from q0.
|
||
let start = predictor.map(|p| self.state(p, lambda, true).map(|e| (p.to_vec(), e)));
|
||
let (mut q, mut eval) = match start {
|
||
Some(Ok(s)) => s,
|
||
Some(Err(Failure::Fatal(e))) => return Err(Failure::Fatal(e)),
|
||
Some(Err(Failure::Cutback(_))) | None => (q0.to_vec(), self.state(q0, lambda, true)?),
|
||
};
|
||
let mut r = self.residual(&eval);
|
||
// With path-dependent materials, an increment that starts within the force tolerance (a
|
||
// predictor that already solved it) has only round-off left for the energy criterion to
|
||
// measure against: it is waived.
|
||
let started_converged = self.path_dependent && self.force_floor.is_some() && norm_inf(&r) <= tol.force * self.scale();
|
||
let prescribed = size * red.largest_prescribed();
|
||
// The energy criterion's reference: the first correction's work, or the tangent
|
||
// predictor's when the increment started from one that is still in use.
|
||
let mut first_energy: Option<f64> = predicted_energy.filter(|_| q.as_slice() != q0);
|
||
let mut history: Vec<f64> = Vec::new();
|
||
let max_iterations = usize::try_from(self.settings.max_iterations).unwrap_or(usize::MAX);
|
||
for _ in 0..max_iterations {
|
||
// The round-off rule: nothing left to correct.
|
||
if norm_inf(&r) <= ROUND_OFF * tol.force * self.scale() && self.force_floor.is_some() {
|
||
return Ok(Converged { q, evaluation: eval });
|
||
}
|
||
if self.iterations >= self.limits.max_newton_iterations {
|
||
return Err(Failure::Fatal(Error::Limit(format!(
|
||
"the step took {} Newton iterations, the limit (raise max_newton_iterations or loosen the tolerances)",
|
||
self.iterations
|
||
))));
|
||
}
|
||
self.iterations += 1;
|
||
let k = eval.tangent.take().ok_or_else(|| Failure::Fatal(Error::Internal("tangent".into())))?;
|
||
let [kff] = red.reduce([&k])?;
|
||
// The first tangent of the step: a singular one is a modelling error, not a cutback.
|
||
let first = self.force_floor.is_none();
|
||
if first {
|
||
let diagonal = kff.diagonal();
|
||
let mean = diagonal.iter().map(|d| d.abs()).sum::<f64>() / diagonal.len().max(1) as f64;
|
||
self.force_floor = Some(1e-8 * mean * self.length);
|
||
}
|
||
drop(k);
|
||
let chol = match factor_raw(&kff, self.limits) {
|
||
Ok(c) => c,
|
||
Err(FactorError::NotPositiveDefinite { .. }) if !first || self.undeformed_is_stable()? => {
|
||
return Err(Failure::Cutback(
|
||
"the tangent stiffness is not positive definite (a limit point, a bifurcation or an unstable material)".into(),
|
||
));
|
||
}
|
||
Err(e) => return Err(Failure::Fatal(factor_error(e, self.problem.mesh, red))),
|
||
};
|
||
self.factor_entries = chol.entries();
|
||
drop(kff);
|
||
let mut dq = r.clone();
|
||
chol.solve(&mut dq)?;
|
||
if dq.iter().any(|v| !v.is_finite()) {
|
||
return Err(Failure::Cutback("the Newton correction is not finite".into()));
|
||
}
|
||
let g0 = dot(&dq, &r);
|
||
let (s, next) = self.line_search(&q, &dq, lambda, g0)?;
|
||
for (x, d) in q.iter_mut().zip(&dq) {
|
||
*x += s * d;
|
||
}
|
||
eval = next;
|
||
let r_old = std::mem::replace(&mut r, self.residual(&eval));
|
||
let energy = (s * dot(&dq, &r_old)).abs();
|
||
let first = *first_energy.get_or_insert(energy);
|
||
let moved = q.iter().zip(q0).fold(0.0f64, |m, (a, b)| m.max((a - b).abs()));
|
||
let reference = moved.max(prescribed).max(1e-12 * self.length);
|
||
let it = Iteration {
|
||
force: norm_inf(&r) / self.scale(),
|
||
displacement: s * norm_inf(&dq) / reference,
|
||
energy: if first > 0.0 { energy / first } else { 0.0 },
|
||
line_search: s,
|
||
};
|
||
iterations.push(it);
|
||
if !it.force.is_finite() {
|
||
return Err(Failure::Cutback("the residual is not finite".into()));
|
||
}
|
||
let energy_met = it.energy <= tol.energy || started_converged;
|
||
if (it.force <= tol.force && it.displacement <= tol.displacement && energy_met) || it.force <= ROUND_OFF * tol.force {
|
||
return Ok(Converged { q, evaluation: eval });
|
||
}
|
||
history.push(it.force);
|
||
if let [.., a, b, c] = history[..]
|
||
&& c > b
|
||
&& b > a
|
||
{
|
||
return Err(Failure::Cutback(format!("the residual grows ({a:.2e}, {b:.2e}, {c:.2e} of the force scale)")));
|
||
}
|
||
}
|
||
Err(Failure::Cutback(format!("no convergence in {} iterations", self.settings.max_iterations)))
|
||
}
|
||
|
||
/// The line search along `dq` from `q`: the accepted step and the state there (with the
|
||
/// tangent).
|
||
fn line_search(&mut self, q: &[f64], dq: &[f64], lambda: f64, g0: f64) -> std::result::Result<(f64, Evaluation), Failure> {
|
||
let at = |s: f64| -> Vec<f64> { q.iter().zip(dq).map(|(a, b)| a + s * b).collect() };
|
||
// The full step, with its tangent (taken in most iterations).
|
||
let full = self.state(&at(1.0), lambda, true);
|
||
if !self.settings.line_search {
|
||
return full.map(|e| (1.0, e));
|
||
}
|
||
let red = self.problem.red;
|
||
let work = |e: &Evaluation| dot(dq, &free_residual(red, e));
|
||
let mut s = 1.0;
|
||
let mut g = match &full {
|
||
Ok(e) => {
|
||
let g1 = work(e);
|
||
if g1.abs() <= LINE_SEARCH_ACCEPT * g0.abs() {
|
||
return full.map(|e| (1.0, e));
|
||
}
|
||
Some(g1)
|
||
}
|
||
Err(Failure::Cutback(_)) => None,
|
||
Err(Failure::Fatal(_)) => return full.map(|e| (1.0, e)),
|
||
};
|
||
let mut best: Option<(f64, f64)> = g.map(|g| (1.0, g.abs()));
|
||
for _ in 0..LINE_SEARCH_TRIALS {
|
||
// Root of the line through (0, g0) and (s, g), or half the step after a failure.
|
||
let next = match g {
|
||
Some(gs) if (g0 - gs).abs() > 0.0 => (s * g0 / (g0 - gs)).clamp(LINE_SEARCH_MIN, 1.0),
|
||
Some(_) => break,
|
||
None => (0.5 * s).max(LINE_SEARCH_MIN),
|
||
};
|
||
if next == s {
|
||
break;
|
||
}
|
||
s = next;
|
||
g = match self.state(&at(s), lambda, false) {
|
||
Ok(e) => Some(work(&e)),
|
||
Err(Failure::Cutback(_)) => None,
|
||
Err(f) => return Err(f),
|
||
};
|
||
if let Some(gs) = g {
|
||
if best.is_none_or(|(_, b)| gs.abs() < b) {
|
||
best = Some((s, gs.abs()));
|
||
}
|
||
if gs.abs() <= LINE_SEARCH_ACCEPT * g0.abs() {
|
||
break;
|
||
}
|
||
}
|
||
}
|
||
let Some((s, _)) = best else {
|
||
return Err(Failure::Cutback("every step along the Newton correction inverts an element".into()));
|
||
};
|
||
if s == 1.0 {
|
||
return full.map(|e| (1.0, e));
|
||
}
|
||
self.state(&at(s), lambda, true).map(|e| (s, e))
|
||
}
|
||
}
|
||
|
||
/// Refuse what nonlinear steps cannot take yet, and warn about what they simplify.
|
||
fn check_supported(model: &Model, mesh: &Mesh, step: &Step, settings: &NonlinearStatic, warnings: &mut Vec<String>) -> Result<()> {
|
||
let at = |what: String| Error::NotSupported(format!("{what} in nonlinear steps (step `{}`)", step.name));
|
||
if let Some(t) = mesh.trusses.first() {
|
||
return Err(at(format!("trusses (element {})", t.label)));
|
||
}
|
||
if let Some(s) = mesh.structural.first() {
|
||
return Err(at(format!("shells and beams, whose large rotations are not implemented yet (element {})", s.label)));
|
||
}
|
||
for e in &mesh.elements {
|
||
if !settings.nonlinear_geometry && e.hyperelastic.is_some() {
|
||
return Err(at(format!("a hyperelastic material without nonlinear geometry (element {})", e.label)));
|
||
}
|
||
}
|
||
if let Some(name) = step.loads.iter().find(|n| matches!(model.load(n), Some(Load::Centrifugal { .. }))) {
|
||
return Err(at(format!("centrifugal loads (load `{name}`), which depend on the deformed position")));
|
||
}
|
||
plasticity(model, step, settings, warnings)?;
|
||
if settings.nonlinear_geometry
|
||
&& let Some(name) = model.couplings().keys().next()
|
||
{
|
||
warnings.push(format!(
|
||
"coupling `{name}` and the others are linear constraints (small rotations of their reference points) in this nonlinear step"
|
||
));
|
||
}
|
||
Ok(())
|
||
}
|
||
|
||
/// Refuse the plasticity a static step cannot take, and warn about what it leaves out, for the
|
||
/// materials of the solid and 2D sections.
|
||
fn plasticity(model: &Model, step: &Step, settings: &NonlinearStatic, warnings: &mut Vec<String>) -> Result<()> {
|
||
let at = |what: String| Error::NotSupported(format!("{what} in nonlinear steps (step `{}`)", step.name));
|
||
let used: std::collections::BTreeSet<&str> = model
|
||
.sections()
|
||
.values()
|
||
.filter(|s| matches!(s.kind, SectionKind::Solid {}) || Theory::from_section(&s.kind).is_some())
|
||
.map(|s| s.material.as_str())
|
||
.collect();
|
||
for name in used {
|
||
let Some(m) = model.material(name) else { continue };
|
||
match &m.plastic {
|
||
Some(Plastic::JohnsonCook { c, specific_heat, .. }) => {
|
||
if *c != 0.0 {
|
||
return Err(at(format!(
|
||
"the Johnson–Cook strain-rate term of material `{name}` (C = {c}): a static step has no time, so no strain rate; set C to 0 or use explicit dynamics"
|
||
)));
|
||
}
|
||
if specific_heat.is_some() {
|
||
warnings.push(format!(
|
||
"material `{name}`: a static step is isothermal, so plastic work does not heat it (adiabatic heating acts in explicit dynamics)"
|
||
));
|
||
}
|
||
}
|
||
Some(Plastic::Bilinear { hardening: Hardening::Kinematic, .. }) if settings.nonlinear_geometry => {
|
||
return Err(at(format!(
|
||
"the kinematic hardening of material `{name}` at finite strain (its back stress is not coaxial with the elastic stretch, which the exponential return map needs); use nonlinear_geometry false or isotropic hardening"
|
||
)));
|
||
}
|
||
_ => {}
|
||
}
|
||
if m.plastic.is_some() && (m.damage.is_some() || m.erosion.is_some()) {
|
||
warnings.push(format!("material `{name}`: damage and erosion act in explicit dynamics; this static step leaves them out"));
|
||
}
|
||
}
|
||
Ok(())
|
||
}
|
||
|
||
/// The step's temperature field per mesh node: the steady heat solution of its temperature
|
||
/// boundary conditions and heat loads when it has some (one-way coupling, as the coupled
|
||
/// temperature–displacement step of the linear solver), or its temperature loads.
|
||
fn temperatures(model: &Model, resolved: &fieldcraft_model::Resolved, mesh: &Mesh, step: &Step, limits: &Limits) -> Result<Option<Vec<Option<f64>>>> {
|
||
let conducts = step.boundary_conditions.iter().any(|b| model.boundary_conditions().get(b).is_some_and(BoundaryCondition::is_thermal))
|
||
|| step.loads.iter().any(|l| model.load(l).is_some_and(Load::is_thermal));
|
||
let loads = temperature_loads(model, resolved, mesh, step)?;
|
||
match (conducts, loads) {
|
||
(true, Some(_)) => Err(Error::Invalid(format!(
|
||
"step `{}` gives temperatures both by temperature loads and by heat conditions (temperature boundary conditions, heat loads); use one",
|
||
step.name
|
||
))),
|
||
(true, None) => Ok(Some(crate::heat::temperatures_at(model, resolved, step, limits, &mesh.labels)?)),
|
||
(false, field) => Ok(field),
|
||
}
|
||
}
|
||
|
||
/// The follower pressures of a step (with nonlinear geometry), and the step without them for
|
||
/// the dead loads.
|
||
fn followers(model: &Model, resolved: &fieldcraft_model::Resolved, mesh: &Mesh, step: &Step) -> Result<(Vec<Follower>, Step)> {
|
||
let mut out = Vec::new();
|
||
let mut dead = step.clone();
|
||
for name in &step.loads {
|
||
let Some(Load::Pressure { set, magnitude }) = model.load(name) else { continue };
|
||
dead.loads.remove(name);
|
||
let slots = |labels: Vec<u32>| -> Result<(Vec<usize>, Vec<[f64; 3]>, Vec<usize>)> {
|
||
let idx: Vec<usize> = labels
|
||
.iter()
|
||
.map(|l| mesh.node(*l).ok_or_else(|| Error::Invalid(format!("load `{name}`: node {l} belongs to no solid element"))))
|
||
.collect::<Result<_>>()?;
|
||
let xyz = idx.iter().filter_map(|i| mesh.xyz.get(*i).copied()).collect();
|
||
let mut dofs = Vec::with_capacity(3 * idx.len());
|
||
for n in &idx {
|
||
dofs.extend(translations(mesh, *n)?);
|
||
}
|
||
Ok((idx, xyz, dofs))
|
||
};
|
||
if let Some(theory) = mesh.theory {
|
||
if resolved.sets.get(set).is_some_and(|s| !s.faces.is_empty()) {
|
||
return Err(Error::Invalid(format!("load `{name}`: a 2D model's pressures act on element edges, and set `{set}` has element faces")));
|
||
}
|
||
for g in resolved.planar_edges(set) {
|
||
let labels =
|
||
model.edge_nodes(g).ok_or_else(|| Error::Invalid(format!("load `{name}`: element {} edge {} is missing", g.element, g.edge)))?;
|
||
let (nodes, xyz, dofs) = slots(labels)?;
|
||
out.push(Follower { element: g.element, nodes, xyz, magnitude: *magnitude, extent: Some(theory.extent()), dofs });
|
||
}
|
||
} else {
|
||
for fc in resolved.sets.get(set).map(|s| s.faces.as_slice()).unwrap_or_default() {
|
||
let labels = model
|
||
.face_nodes(*fc)
|
||
.ok_or_else(|| Error::Invalid(format!("load `{name}`: element {} face {} is missing", fc.element, fc.face)))?;
|
||
let (nodes, xyz, dofs) = slots(labels)?;
|
||
out.push(Follower { element: fc.element, nodes, xyz, magnitude: *magnitude, extent: None, dofs });
|
||
}
|
||
}
|
||
}
|
||
Ok((out, dead))
|
||
}
|
||
|
||
/// Solve a nonlinear static step: the step named `step`, or the model's only nonlinear static
|
||
/// step.
|
||
pub fn solve_nonlinear(model: &Model, step: Option<&str>, limits: &Limits) -> Result<StaticResult> {
|
||
let resolved = model.resolve()?;
|
||
let step = model.select_step(step, |p| matches!(p, Procedure::NonlinearStatic(_)), "nonlinear static")?;
|
||
let Procedure::NonlinearStatic(settings) = &step.procedure else {
|
||
return Err(Error::Internal("nonlinear static step".into()));
|
||
};
|
||
settings.validate()?;
|
||
let mesh = Mesh::new(model, &resolved, limits.max_dofs)?;
|
||
let mut warnings: Vec<String> = resolved.warnings.iter().map(ToString::to_string).chain(mesh.warnings.iter().cloned()).collect();
|
||
check_supported(model, &mesh, step, settings, &mut warnings)?;
|
||
let (graph, order) = pattern(&mesh);
|
||
ignored_dashpots(&mesh, &mut warnings);
|
||
let setup = Setup { model, resolved: &resolved, mesh: &mesh, step };
|
||
let red = reduction(&setup, true, &order, budget(limits, &mesh, &graph, 1), &mut warnings)?;
|
||
if red.neq() == 0 {
|
||
return Err(Error::Invalid("every degree of freedom is prescribed; there is nothing to solve".into()));
|
||
}
|
||
let (follower_loads, dead_step) =
|
||
if settings.nonlinear_geometry { followers(model, &resolved, &mesh, step)? } else { (Vec::new(), step.clone()) };
|
||
let dead = loads(model, &resolved, &mesh, &dead_step, &red.active)?;
|
||
let field = temperatures(model, &resolved, &mesh, step, limits)?;
|
||
let springs = if mesh.discrete.springs.iter().any(|s| !s.dashpot) {
|
||
let mut s = graph.zero_matrix(&mesh.layout);
|
||
mesh.discrete.add_stiffness(&mut s)?;
|
||
Some(s)
|
||
} else {
|
||
None
|
||
};
|
||
let element_kinds = mesh.elements.iter().map(|e| Ok(e.kernel()?.dofs().node)).collect::<Result<Vec<_>>>()?;
|
||
let element_dofs = mesh
|
||
.elements
|
||
.iter()
|
||
.zip(&element_kinds)
|
||
.map(|(e, kinds)| {
|
||
mesh.layout.element_dofs(&e.nodes, *kinds).ok_or_else(|| Error::Internal(format!("element {} degrees of freedom", e.label)))
|
||
})
|
||
.collect::<Result<Vec<_>>>()?;
|
||
let mut problem = Problem {
|
||
mesh: &mesh,
|
||
red: &red,
|
||
dead,
|
||
followers: follower_loads,
|
||
springs,
|
||
finite: settings.nonlinear_geometry,
|
||
element_dofs,
|
||
element_kinds,
|
||
materials: mesh
|
||
.elements
|
||
.iter()
|
||
.map(|e| match &e.plastic {
|
||
Some(j2) => MaterialModel::Plastic(j2.clone()),
|
||
None => MaterialModel::from_elasticity(&e.d, e.hyperelastic.as_ref()),
|
||
})
|
||
.collect(),
|
||
field,
|
||
pattern: graph.zero_matrix(&mesh.layout),
|
||
};
|
||
// Elements on their linear stiffness take the thermal strain as the equivalent nodal
|
||
// forces, a dead load proportional to the load factor like the temperatures.
|
||
if let Some(field) = problem.field.take() {
|
||
let linear: Vec<bool> = (0..mesh.elements.len()).map(|i| problem.kinematics(i).is_none()).collect();
|
||
add_element_thermal_loads(&mut problem.dead, &mesh, &field, |i| linear.get(i).copied().unwrap_or(false))?;
|
||
problem.field = Some(field);
|
||
}
|
||
let (lo, hi) = mesh.xyz.iter().fold(([f64::INFINITY; 3], [f64::NEG_INFINITY; 3]), |(lo, hi), p| {
|
||
(std::array::from_fn(|k| lo[k].min(p[k])), std::array::from_fn(|k| hi[k].max(p[k])))
|
||
});
|
||
let length = (0..3).map(|k| (hi[k] - lo[k]).powi(2)).sum::<f64>().sqrt().max(f64::MIN_POSITIVE);
|
||
let committed = mesh
|
||
.elements
|
||
.iter()
|
||
.zip(&problem.materials)
|
||
.map(|(e, m)| Ok(vec![0.0; e.kernel()?.rule().points().len() * m.history_len()]))
|
||
.collect::<Result<Vec<_>>>()?;
|
||
// The undeformed state first: an element that is bad before anything moves is an error in
|
||
// the model, not a reason to cut increments back.
|
||
problem.evaluate(&vec![0.0; mesh.dofs()], 0.0, false, &committed)?;
|
||
let path_dependent = problem.materials.iter().any(MaterialModel::is_path_dependent);
|
||
let mut run = Run {
|
||
problem: &problem,
|
||
settings,
|
||
limits,
|
||
force_scale: 0.0,
|
||
force_floor: None,
|
||
length,
|
||
iterations: 0,
|
||
factor_entries: 0,
|
||
committed,
|
||
path_dependent,
|
||
undeformed_stable: None,
|
||
};
|
||
|
||
let mut history = History::default();
|
||
let (mut frame_values, mut frames_full) = (0usize, false);
|
||
let values = 3 * mesh.labels.len();
|
||
let mut keep_frame = |history: &mut History, lambda: f64, u: &[f64], warnings: &mut Vec<String>| {
|
||
if frames_full {
|
||
return;
|
||
}
|
||
if frame_values.saturating_add(values) > MAX_FRAME_VALUES {
|
||
warnings.push(format!("the displacement frames stop at load factor {lambda}: the step keeps at most {MAX_FRAME_VALUES} values"));
|
||
frames_full = true;
|
||
return;
|
||
}
|
||
frame_values += values;
|
||
history.frames.push(Frame { load_factor: lambda, displacement: split(u, &mesh).0 });
|
||
};
|
||
keep_frame(&mut history, 0.0, &vec![0.0; mesh.dofs()], &mut warnings);
|
||
let mut q = vec![0.0; red.neq()];
|
||
// The step fraction t and the load factor λ(t) of the step's load path (λ = t without one).
|
||
let (mut t, mut lambda) = (0.0f64, 0.0f64);
|
||
let mut size = settings.initial_increment;
|
||
let mut fast = 0usize;
|
||
let mut last: Option<Evaluation> = None;
|
||
// The last converged increment's free motion and the change of the load factor over it.
|
||
let mut previous: Option<(Vec<f64>, f64)> = None;
|
||
let max_increments = usize::try_from(settings.max_increments).unwrap_or(usize::MAX);
|
||
while t < 1.0 {
|
||
if history.increments.len() >= max_increments {
|
||
return Err(Error::Limit(format!(
|
||
"step `{}`: {} increments reached the step's cap at load factor {lambda:.6} (raise max_increments)",
|
||
step.name, settings.max_increments
|
||
)));
|
||
}
|
||
// An increment ends at the next point of the load path at the latest.
|
||
let turn = settings.next_turn(t);
|
||
let remaining = turn - t;
|
||
let (target, this) = if size >= remaining * (1.0 - 1e-9) { (turn, remaining) } else { (t + size, size) };
|
||
let target_lambda = settings.load_factor(target);
|
||
let change = if settings.load_path.is_some() { target_lambda - lambda } else { this };
|
||
let mut iterations = Vec::new();
|
||
let number = history.increments.len() + 1;
|
||
// Linear extrapolation of the last converged increment's free motion, in proportion to
|
||
// the change of the load factor (so it turns with the load path).
|
||
let mut predictor: Option<Vec<f64>> = previous
|
||
.as_ref()
|
||
.filter(|(_, prev)| *prev != 0.0)
|
||
.map(|(dq, prev): &(Vec<f64>, f64)| q.iter().zip(dq).map(|(a, d)| a + change / prev * d).collect());
|
||
// Path-dependent materials start without one from the tangent predictor.
|
||
let mut predicted_energy = None;
|
||
if predictor.is_none() && path_dependent {
|
||
match run.tangent_predictor(&q, lambda, target_lambda) {
|
||
Ok(Some((p, energy))) => (predictor, predicted_energy) = (Some(p), Some(energy)),
|
||
Ok(None) | Err(Failure::Cutback(_)) => {}
|
||
Err(Failure::Fatal(e)) => return Err(e),
|
||
}
|
||
}
|
||
match run.increment(&q, predictor.as_deref(), predicted_energy, target_lambda, change.abs(), &mut iterations) {
|
||
Ok(c) => {
|
||
previous = Some((c.q.iter().zip(&q).map(|(a, b)| a - b).collect(), change));
|
||
let count = iterations.len();
|
||
history.increments.push(Increment { number, load_factor: target_lambda, size: this, iterations, converged: true, cutback: None });
|
||
(t, lambda) = (target, target_lambda);
|
||
q = c.q;
|
||
keep_frame(&mut history, lambda, &red.expand_scaled(&q, lambda), &mut warnings);
|
||
run.committed.clone_from(&c.evaluation.history);
|
||
last = Some(c.evaluation);
|
||
fast = if count <= FAST_ITERATIONS { fast + 1 } else { 0 };
|
||
if fast >= 2 {
|
||
size *= GROWTH;
|
||
} else if count > SLOW_ITERATIONS {
|
||
size *= SHRINK;
|
||
}
|
||
size = size.clamp(settings.min_increment, settings.max_increment);
|
||
}
|
||
Err(Failure::Fatal(e)) => return Err(e),
|
||
Err(Failure::Cutback(reason)) => {
|
||
history.increments.push(Increment {
|
||
number,
|
||
load_factor: target_lambda,
|
||
size: this,
|
||
iterations,
|
||
converged: false,
|
||
cutback: Some(reason.clone()),
|
||
});
|
||
fast = 0;
|
||
size = this * CUTBACK;
|
||
if size < settings.min_increment * (1.0 - 1e-9) {
|
||
return Err(Error::NotConverged(format!(
|
||
"step `{}` stopped at load factor {lambda:.6} (step fraction {t:.6}): the increment to {target_lambda:.6} failed ({reason}) and a smaller one would be below the minimum increment {}",
|
||
step.name, settings.min_increment
|
||
)));
|
||
}
|
||
}
|
||
}
|
||
}
|
||
history.iterations = run.iterations;
|
||
let u = red.expand_scaled(&q, lambda);
|
||
let evaluation = match last {
|
||
Some(e) => e,
|
||
None => problem.evaluate(&u, lambda, false, &run.committed)?,
|
||
};
|
||
if let Some(a) = problem.follower_asymmetry(&u, lambda)?
|
||
&& a > 1e-6
|
||
{
|
||
warnings.push(format!(
|
||
"follower pressure: the skew part of its load stiffness ({a:.1e} of it, over the free degrees of freedom) is left out of the tangent; the solution is exact, Newton may converge more slowly"
|
||
));
|
||
}
|
||
finish(&mesh, step, &red, &problem, u, lambda, evaluation, &run.committed, history, run.factor_entries, warnings)
|
||
}
|
||
|
||
/// The final state as a static result: displacements, reactions, Cauchy stresses and
|
||
/// logarithmic strains.
|
||
#[allow(clippy::too_many_arguments)]
|
||
fn finish(
|
||
mesh: &Mesh,
|
||
step: &Step,
|
||
red: &Reduction,
|
||
problem: &Problem,
|
||
u: Vec<f64>,
|
||
lambda: f64,
|
||
e: Evaluation,
|
||
committed: &[Vec<f64>],
|
||
history: History,
|
||
factor_entries: usize,
|
||
mut warnings: Vec<String>,
|
||
) -> Result<StaticResult> {
|
||
let (displacement, rotation, point_motion) = split(&u, mesh);
|
||
let out_of_balance: Vec<f64> = e.internal.iter().zip(&e.external).map(|(a, b)| a - b).collect();
|
||
let reactions = red.reactions_of(&out_of_balance);
|
||
let (reaction, reaction_moment, point_reaction) = split(&reactions, mesh);
|
||
let (mut applied_total, mut reaction_total) = ([0.0; 3], [0.0; 3]);
|
||
for (dof, (fv, rv)) in e.external.iter().zip(&reactions).enumerate() {
|
||
let Some(d) = mesh.kind(dof).and_then(DofKind::direction) else { continue };
|
||
if let (Some(a), Some(r)) = (applied_total.get_mut(d), reaction_total.get_mut(d)) {
|
||
*a += fv;
|
||
*r += rv;
|
||
}
|
||
}
|
||
let reference_points = mesh
|
||
.points
|
||
.iter()
|
||
.zip(point_motion.iter().zip(&point_reaction))
|
||
.map(|(p, (d, r))| PointResult { name: p.name.clone(), label: p.label, displacement: *d, reaction: *r })
|
||
.collect();
|
||
#[allow(clippy::type_complexity)]
|
||
let all_points = parallel::map_grained(mesh.elements.len(), GRAIN, |i| -> Result<(Vec<PointStress>, Vec<f64>, &'static ElementType, Rule)> {
|
||
let el = mesh.elements.get(i).ok_or_else(|| Error::Internal("element index".into()))?;
|
||
let at = |err| Error::element(el.label, err);
|
||
let ue = problem.element_vector(i, &u);
|
||
let kernel = el.kernel()?;
|
||
let history = committed.get(i).map(Vec::as_slice).unwrap_or_default();
|
||
let material = problem.materials.get(i).ok_or_else(|| Error::Internal("element material".into()))?;
|
||
let points = match problem.kinematics(i) {
|
||
Some(kin) => kernel.nonlinear_stresses(material, &ue, kin, problem.temperature(i, lambda).as_ref(), history),
|
||
None => {
|
||
let per_node: Vec<[f64; 3]> = el.nodes.iter().map(|n| displacement.get(*n).copied().unwrap_or_default()).collect();
|
||
let thermal = problem.field.as_deref().and_then(|f| el.thermal_strain(f)).map(|mut t| {
|
||
t.delta.iter_mut().for_each(|d| *d *= lambda);
|
||
t
|
||
});
|
||
kernel.stresses_with(&el.d, &per_node, thermal.as_ref())
|
||
}
|
||
}
|
||
.map_err(at)?;
|
||
let len = material.history_len();
|
||
let plastic: Vec<f64> = if len > 0 { history.chunks(len).filter_map(|h| material.plastic_strain(h)).collect() } else { Vec::new() };
|
||
Ok((points, plastic, kernel.ty(), kernel.rule()))
|
||
});
|
||
let mut extrapolations: Vec<(&'static ElementType, Rule, Extrapolation)> = Vec::new();
|
||
let mut elements = Vec::with_capacity(mesh.elements.len());
|
||
let mut sum = vec![[0.0; 6]; mesh.labels.len()];
|
||
let mut count = vec![0u32; mesh.labels.len()];
|
||
for (el, computed) in mesh.elements.iter().zip(all_points) {
|
||
let (points, plastic_strain, ty, rule) = computed?;
|
||
let values: Vec<[f64; 6]> = points.iter().map(|p| p.stress).collect();
|
||
let index = match extrapolations.iter().position(|(t, r, _)| std::ptr::eq(*t, ty) && *r == rule) {
|
||
Some(i) => i,
|
||
None => {
|
||
extrapolations.push((ty, rule, extrapolation(ty, rule).map_err(|err| Error::element(el.label, err))?));
|
||
extrapolations.len() - 1
|
||
}
|
||
};
|
||
let nodal = extrapolations.get(index).map(|(_, _, x)| x.apply(&values)).unwrap_or_default();
|
||
for (n, s) in el.nodes.iter().zip(&nodal) {
|
||
if let (Some(acc), Some(c)) = (sum.get_mut(*n), count.get_mut(*n)) {
|
||
for k in 0..6 {
|
||
acc[k] += s[k];
|
||
}
|
||
*c += 1;
|
||
}
|
||
}
|
||
elements.push(ElementStress { label: el.label, kind: el.kind, points, nodal, plastic_strain });
|
||
}
|
||
let stress: Vec<[f64; 6]> = sum.iter().zip(&count).map(|(s, c)| s.map(|v| if *c > 0 { v / f64::from(*c) } else { 0.0 })).collect();
|
||
let von_mises = stress.iter().map(von_mises).collect();
|
||
let scale = applied_total.iter().chain(&reaction_total).fold(0.0f64, |m, v| m.max(v.abs()));
|
||
let ground = mesh.discrete.ground_force(&u, mesh);
|
||
let imbalance = (0..3).map(|c| (applied_total[c] + reaction_total[c] + ground[c]).abs()).fold(0.0, f64::max);
|
||
if scale > 0.0 && imbalance > 1e-6 * scale {
|
||
warnings.push(format!("loads and reactions differ by {imbalance:.3e} (relative {:.1e})", imbalance / scale));
|
||
}
|
||
Ok(StaticResult {
|
||
step: step.name.clone(),
|
||
nodes: mesh.labels.clone(),
|
||
displacement,
|
||
rotation,
|
||
reaction,
|
||
reaction_moment,
|
||
stress,
|
||
von_mises,
|
||
elements,
|
||
shells: Vec::new(),
|
||
beams: Vec::new(),
|
||
reference_points,
|
||
applied_total,
|
||
reaction_total,
|
||
stats: Stats { dofs: mesh.dofs(), equations: red.neq(), factor_entries, lanczos_vectors: 0 },
|
||
warnings,
|
||
temperature: None,
|
||
nonlinear: Some(history),
|
||
})
|
||
}
|